Beijing Institute of Mathematical Sciences and Applications

Semen Artamonov , Pavel Nikitin , Shamil Shakirov

Anton Selemenchuk

Friday, January 24, 2025 1:00 PM - 2:30 PM

A14-201

Zoom 242 742 6089 (BIMSA)

Consider an $n\times k$ matrix of i.i.d. Bernoulli random numbers with some value of $p$. Dual RSK algorithm gives a bijection of this matrix to a pair of Young tableaux of conjugate shape, which is manifestation of skew Howe $G{L}_n\times G{L}_k$-duality. Thus the probability measure on zero-ones matrix leads to the probability measure on Young diagrams proportional to the ratio of the dimension of $G{L}_n\times G{L}_k$-representation and the dimension of the exterior algebra $\bigwedge (C^n\otimes C^k)$. Similarly, by applying Proctor's algorithm based on Berele's modification of the Schensted insertion, we get skew Howe duality for the pairs of groups $S{p}_{2n}\times S{p}_{2k}$. In the limit when $n,k\to \mathrm{\infty }$, $GL$-case is relatively easily studied by use of free-fermionic representation for the correlation kernel. But for the symplectic groups there is no convenient free-fermionic representation. We use Christoffel transformation to obtain the semiclassical orthogonal polynomials for $S{p}_{2n}\times S{p}_{2k}$.